All elementary particles belong to one of two families: fermions or bosons. Electrons are fermions; photons are bosons. Things are different, however, for the elementary excitations that emerge in two-dimensional quantum many-body systems—the so-called “quasi-particles.” The latter may belong to infinitely many families with “traits” in-between the fermionic and the bosonic. Such quasiparticles are also known as anyons. Even more fundamentally interesting is the prediction that some anyons are non-Abelian. This means that a system of many such anyons has a large number of degenerate ground states and can be in any of those or their superpositions, depending in an intricate manner on the history of the system. Where may these exotic particles be discovered? The most prominently studied candidate is a two-dimensional electron gas in a perpendicular magnetic field of proper strength such that the system shows the so-called fractional quantum Hall effect. It turns out that, because of the complex nature of the underlying states determined by electron-electron interaction, finding out precisely what the unique “traits” (or “statistics”) are of non-Ablian anyons is challenging. The theoretical machinery used so far to tackle this problem is both technically heavy, involving conformal field theory, and rests on some assumptions that call for independent confirmation. Among the most complicated states that have been conjectured to have an experimental realization, independently confirmed results have been available thus far only for the Moore-Read or “Pfaffian” state. In this theoretical paper, we present a new method that makes no use of conformal field theory, and demonstrate its application not only to the Moore-Read state, but also to the so-called $k=3$ Read-Rezayi state, another state that is notoriously difficult to tackle.

Our method grows out of a strategy that has been successfully applied to interacting many-body systems since the concept of a Fermi liquid was proposed by Landau in 1956, but only recently to systems of interacting electrons in the fractional quantum Hall regime. It views a complicated state of many interacting electrons as the descendant of a simple fictitious system of noninteracting electrons, following a particular rule of evolution (“adiabatic continuity”). Our work achieves a number of things. It leads to an entirely new description of quasiparticles in the $k=3$ Read-Rezayi state. More generally, it provides a framework for calculation that leads efficiently from data extracted from a wave function to the statistics. Our framework is relatively easy to use: Anyone with standard training in theoretical condensed matter physics should be able to pick it up and apply it. It also gives a simple, intuitive graphical representation of non-Abelian statistics. And finally, our method provides an independent alternative that can support the conformal-field-theory-based approach.

We believe that this work represents a significant advance in theoretical condensed matter physics.